Find the series development in z of $g(z) = \frac{-2z - 22}{(z+2)(z-4)}$ and give the radius of convergence

Question

So, the first thing I did was find the partial fractions which are

$g(z) = \frac{3}{z+2} + \frac{-5}{z-4}$

When I'm in $|z| < 2$ I can expand them both in Taylor series, if I'm in $|z| < 4$ I can expand the first one as a Laurent series and the second one as a Taylor series and if I'm in $|z| > 4$ I have to expand them both as a Laurent series. Is that right?